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 FresnelC

 http://functions.wolfram.com/06.33.06.0014.01

 Input Form

 FresnelC[z] == Pi^(3/2) Sum[((2^(2 k - 3/2) Subscript[z, 0]^(1 - k))/k!) HypergeometricPFQRegularized[{1/4, 3/4, 1}, {(2 - k)/4, (3 - k)/4, 1 - k/4, (5 - k)/4}, -((Pi^2 Subscript[z, 0]^4)/16)] (z - Subscript[z, 0])^k, {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["FresnelC", "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", FractionBox["3", "2"]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["1", "-", "k"]]]]], RowBox[List["k", "!"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["3", "4"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "-", "k"]], "4"], ",", FractionBox[RowBox[List["3", "-", "k"]], "4"], ",", RowBox[List["1", "-", FractionBox["k", "4"]]], ",", FractionBox[RowBox[List["5", "-", "k"]], "4"]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SubsuperscriptBox["z", "0", "4"]]], "16"]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]

 MathML Form

 C TagBox["C", FresnelC] ( z ) π 3 / 2 k = 0 2 2 k - 3 2 z 0 1 - k k ! 3 F ~ 4 ( 1 4 , 3 4 , 1 ; 2 - k 4 , 3 - k 4 , 1 - k 4 , 5 - k 4 ; - π 2 z 0 4 16 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "4"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["2", "-", "k"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["3", "-", "k"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["k", "4"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["5", "-", "k"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SubsuperscriptBox["z", "0", "4"]]], "16"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ( z - z 0 ) k FresnelC z 3 2 k 0 2 2 k -1 3 2 Subscript z 0 1 -1 k k -1 HypergeometricPFQRegularized 1 4 3 4 1 2 -1 k 4 -1 3 -1 k 4 -1 1 -1 k 4 -1 5 -1 k 4 -1 -1 2 Subscript z 0 4 16 -1 z -1 Subscript z 0 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FresnelC", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", FractionBox["3", "2"]]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["1", "-", "k"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["3", "4"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "-", "k"]], "4"], ",", FractionBox[RowBox[List["3", "-", "k"]], "4"], ",", RowBox[List["1", "-", FractionBox["k", "4"]]], ",", FractionBox[RowBox[List["5", "-", "k"]], "4"]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "16"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SubsuperscriptBox["zz", "0", "4"]]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02